enter image description hereIf I have a complex spectrum and I add it to its complex conjugate will I get a zero phase version of the original? Will this also reduce noise? It seems like this is simply phase cancellation.

  • $\begingroup$ So what is your question? $\endgroup$ – Marcus Müller May 15 '16 at 22:37
  • $\begingroup$ Also, systems can have zero phase property, not spectra. I'm not quite sure what exactly you're referring to. $\endgroup$ – Marcus Müller May 15 '16 at 22:38
  • $\begingroup$ And: what is complex value plus it's conjugate? I think you might be missing an obvious point here. $\endgroup$ – Marcus Müller May 15 '16 at 22:39

A signal in $L^2(\mathbb{R}^n)$ can be represented as the linear combination (addition) of sinusoids with different frequencies, phases and amplitudes.

See here: https://www.youtube.com/watch?v=D9ziTuJ3OCw

The Fourier transform calculates the parameters of these sinusoids, where the argument of a complex-valued coefficient is the phase of the sinusoid and the absolute value is the amplitude.

Any sinusoid can be represented as the sum of a symmetric cosine wave plus an anti-symmetric sine wave. The real and imaginary parts of the Fourier transform coefficients are the weights for the cosine wave and the sine wave, respectively.

Therefore the spectrum of the Fourier transform will be complex valued unless all the sinusoids are cosine waves, that is, the phase of all the sinusoids is 0 or $\pi$. This implies that the signal is also symmetric.

  • If you multiply the spectrum by its conjugate, you will get the absolute value of the spectrum, that is, the sinusoid amplitudes.
  • If you add the conjugate to the spectrum, the imaginary parts will cancel out and you are left with the symmetric part of the signal.
  • If you subtract the conjugate from the spectrum, the real parts will cancel out and you are left with the anti-symmetric part of the signal.
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  • $\begingroup$ Thanks for the response @geometrical. Is it possible and/or advisable to use the symmetric part to resynthesis a waveform via an IFFT? It seems that phases would all be zero'd and this has me wondering if the synthesized waveform would be zero centred. Or am I just insane? $\endgroup$ – cixelsyd May 16 '16 at 3:19
  • $\begingroup$ What do you have in mind with `zero-centred'? Do you want the energy of the waveform to fall equally either side of zero? $\endgroup$ – geometrikal May 16 '16 at 4:11
  • $\begingroup$ I'm trying get my audio waveform to display correctly. Its skewed to the point where I'm almost looking down the barrel of the cylinder. I'm wanting it to render to look like a triangle. (it is a spectrum of a triangle wave). It sounds perfect. $\endgroup$ – cixelsyd May 16 '16 at 4:26
  • $\begingroup$ can you post a pic (or link) or any code? I cant quite visualise the problem. :) $\endgroup$ – geometrikal May 16 '16 at 4:37
  • $\begingroup$ I've added the image above. $\endgroup$ – cixelsyd May 16 '16 at 8:03

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